Free loop space

In the mathematical field of topology, a free loop is a variant of the mathematical notion of a loop. Whereas a loop has a distinguished point on it, called a basepoint, a free loop lacks such a distinguished point. Formally, let ${\displaystyle X}$ be a topological space. Then a free loop in ${\displaystyle X}$ is an equivalence class of continuous functions from the circle ${\displaystyle S^{1}}$ to ${\displaystyle X}$. Two loops are equivalent if they differ by a reparameterization of the circle. That is, ${\displaystyle f\sim g}$ if there exists a homeomorphism ${\displaystyle \psi :S^{1}\rightarrow S^{1}}$ such that ${\displaystyle g=f\circ \psi }$.

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